PID controllers are widely used in industry to regulate temperature, pressure, flow rate, chemical composition, speed and practically every other variable for which a measurement exists. FIG. 1 illustrates at 100 an example diagram of a PID controller 102 and example output signals of a process controlled by a PID controller with different tuning parameters. As shown in FIG. 1(a), the purpose of using the PID controller 102 is to make an output y of a process 110 follow a reference value r. To achieve this purpose, the PID controller 102 calculates an error e as the difference between the measured output y and the reference value r and attempts to minimize the error e. The PID controller 102 includes three elements, a proportional element 104, an integral element 106, and a derivative element 108. The three elements produce output values—the proportional, the integral and derivative values, denoted P, I, and D, respectively. These output values can be interpreted in terms of time. P depends on a present error, I depends on the accumulation of past errors, and D is a prediction of future errors, based on a current rate of change.
The traditional basic equation for the PID control scheme is:
      u    ⁡          (      t      )        =                    K        p            ⁢              e        ⁡                  (          t          )                      +                  K        i            ⁢              ∫                              e            ⁡                          (              t              )                                ⁢                      ⅆ                          (              t              )                                            +                  K        d            ⁢              ⅆ                  ⅆ          t                    ⁢              e        ⁡                  (          t          )                    where u(t) is the output of the PID controller 102, Kp, Ki, Kd are tuning parameters, and e(t) is the error.
Example output signals (y) of the process 110 are shown at 112 in FIG. 1(b). Each curve (except the reference signal) represents an output signal for a particular set of tuning parameters. All output signals of the process 110 oscillate around a reference signal in a decaying sinusoid. Eventually, all output signals converge to the reference signal.